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Theorem dvlog2lem 19999
Description: Lemma for dvlog2 20000. (Contributed by Mario Carneiro, 1-Mar-2015.)
Hypothesis
Ref Expression
dvlog2.s  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
Assertion
Ref Expression
dvlog2lem  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )

Proof of Theorem dvlog2lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvlog2.s . . . . 5  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
2 cnxmet 18282 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
3 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
4 1re 8837 . . . . . . 7  |-  1  e.  RR
5 rexr 8877 . . . . . . 7  |-  ( 1  e.  RR  ->  1  e.  RR* )
64, 5ax-mp 8 . . . . . 6  |-  1  e.  RR*
7 blssm 17968 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
82, 3, 6, 7mp3an 1277 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
91, 8eqsstri 3208 . . . 4  |-  S  C_  CC
109sseli 3176 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
113subid1i 9118 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
12 mnfxr 10456 . . . . . . . . . . . 12  |-  -oo  e.  RR*
13 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
14 iocssre 10729 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (  -oo (,] 0 )  C_  RR )
1512, 13, 14mp2an 653 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  RR
1615sseli 3176 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  RR )
1713a1i 10 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  0  e.  RR )
184a1i 10 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR )
19 elioc2 10713 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (
x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) ) )
2012, 13, 19mp2an 653 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) )
2120simp3bi 972 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  0 )
2216, 17, 18, 21lesub2dd 9389 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2311, 22syl5eqbrr 4057 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
24 ax-resscn 8794 . . . . . . . . . . . 12  |-  RR  C_  CC
2515, 24sstri 3188 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  CC
2625sseli 3176 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  CC )
27 eqid 2283 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2827cnmetdval 18280 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1 ( abs 
o.  -  ) x
)  =  ( abs `  ( 1  -  x
) ) )
293, 26, 28sylancr 644 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( abs `  (
1  -  x ) ) )
30 0le1 9297 . . . . . . . . . . . 12  |-  0  <_  1
3130a1i 10 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  ->  0  <_  1 )
3216, 17, 18, 21, 31letrd 8973 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  1 )
3316, 18, 32abssubge0d 11914 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3429, 33eqtrd 2315 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3523, 34breqtrrd 4049 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
36 cnmet 18281 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3736a1i 10 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( Met `  CC ) )
383a1i 10 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  CC )
39 metcl 17897 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
4037, 38, 26, 39syl3anc 1182 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
41 lenlt 8901 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( 1 ( abs 
o.  -  ) x
)  e.  RR )  ->  ( 1  <_ 
( 1 ( abs 
o.  -  ) x
)  <->  -.  ( 1 ( abs  o.  -  ) x )  <  1 ) )
424, 40, 41sylancr 644 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  <_  ( 1 ( abs  o.  -  ) x )  <->  -.  (
1 ( abs  o.  -  ) x )  <  1 ) )
4335, 42mpbid 201 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  ( 1 ( abs 
o.  -  ) x
)  <  1 )
442a1i 10 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
456a1i 10 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR* )
46 elbl2 17950 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  RR* )  /\  ( 1  e.  CC  /\  x  e.  CC ) )  -> 
( x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4744, 45, 38, 26, 46syl22anc 1183 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  (
x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4843, 47mtbird 292 . . . . 5  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 ) )
4948con2i 112 . . . 4  |-  ( x  e.  ( 1 (
ball `  ( abs  o. 
-  ) ) 1 )  ->  -.  x  e.  (  -oo (,] 0
) )
5049, 1eleq2s 2375 . . 3  |-  ( x  e.  S  ->  -.  x  e.  (  -oo (,] 0 ) )
51 eldif 3162 . . 3  |-  ( x  e.  ( CC  \ 
(  -oo (,] 0 ) )  <->  ( x  e.  CC  /\  -.  x  e.  (  -oo (,] 0
) ) )
5210, 50, 51sylanbrc 645 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  (  -oo (,] 0 ) ) )
5352ssriv 3184 1  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   class class class wbr 4023    o. ccom 4693   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   (,]cioc 10657   abscabs 11719   * Metcxmt 16369   Metcme 16370   ballcbl 16371
This theorem is referenced by:  dvlog2  20000  logtayl  20007  logtayl2  20009  efrlim  20264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-xadd 10453  df-ioc 10661  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-xmet 16373  df-met 16374  df-bl 16375
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